The radio k -chromatic number for corona of graphs

Abstract

For a connected simple graph G and a positive integer k ≤diam(G), a radio k-coloring is an assignment f of positive integers (colors) to the vertices of G such that for every pair of distinct vertices u and v in G, |f(u) - f(v)|≥ 1 + k - d(u,v). The span rck(f) of f is maxvϵV(G)f(v). The minimum of {rck(f): f is a radio k-coloring of G} is called the radio k-chromatic number of G and is denoted by rck(G). If d is the diameter of G, then a radio d-coloring is referred as a radio coloring and the radio d-chromatic number as the radio number, rn(G), of G. In this paper, we obtain an upper bound for the radio k-chromatic number of the corona of two graphs G and H for k ≥ 2, and we obtain a necessary condition for this upper bound to be exact. Also, we see that for path Pm, m even and complete graph Kn the upper bound is sharp for rck(Kn - H) and rn(Pm - H), where H is an arbitrary graph. Further, we give a lower bound and an improved upper bound for rn(Pm - H), m odd and rn(Qn - H), Qn is hypercube. © 2023 World Scientific Publishing Company.